Question: Solve for $X$. $\left[\begin{array}{rr}7 & 2 & 1 \\ 9 & 9 & 4 \\0 &1 &-8\end{array}\right]+X=\left[\begin{array}{rr}0 & 3 & -6 \\ 2 & 8 & 1 \\9 &-7 &-5\end{array}\right] $ $X=$
Answer: The Strategy First, we can represent the matrices of the equation with letters, which will make the equation easier to handle. Then we can solve the equation for $X$ and obtain an expression with the letters we defined. Finally, we can substitute back the actual matrices into the resulting expression and simplify it. Solving the equation for $X$ We are given the following equation. $\left[\begin{array}{rr}7 & 2 & 1 \\ 9 & 9 & 4 \\0 &1 &-8\end{array}\right]+X=\left[\begin{array}{rr}0 & 3 & -6 \\ 2 & 8 & 1 \\9 &-7 &-5\end{array}\right] $ Let's represent the above matrices as follows. $A=\left[\begin{array}{rr}7 & 2 & 1 \\ 9 & 9 & 4 \\0 &1 &-8\end{array}\right]~~~~~~~~~ B = \left[\begin{array}{rr}0 & 3 & -6 \\ 2 & 8 & 1 \\9 &-7 &-5\end{array}\right] $ Then we can rewrite the equation as follows. $A+X=B$ Now it's simple to solve the equation for $X$. $\begin{aligned}A+X&=B\\\\ X&=B-A\end{aligned}$ Finding $X$ We found that $X=B-A$. Now we can substitute the actual matrices back into the expression and simplify. $\begin{aligned}X&=B-A \\\\&=\left[\begin{array}{rr}0 & 3 & -6 \\ 2 & 8 & 1 \\9 &-7 &-5\end{array}\right]-\left[\begin{array}{rr}7 & 2 & 1 \\ 9 & 9 & 4 \\0 &1 &-8\end{array}\right] \\\\\\&=\left[\begin{array}{rr}(0-7) & (3-2) & (-6-1) \\ (2-9) & (8-9) & (1-4) \\(9-0) &(-7-1) &(-5+8)\end{array}\right] \\\\\\&=\left[\begin{array}{rr}-7 & 1 & -7 \\ -7 & -1 & -3 \\9 &-8 &3\end{array}\right]\end{aligned}$ Summary $X=\left[\begin{array}{rr}-7 & 1 & -7 \\ -7 & -1 & -3 \\9 &-8 &3\end{array}\right]$